Embodiments of the present invention relate to data processing, and more particularly relate to techniques for decomposing a signal using a sawtooth, or triangle wave, transform.
Many applications in the field of data/signal processing require the decomposition of an input signal (i.e., data set) into one or more constituent frequency components (i.e., component functions). This decomposition is typically performed using a transform, such as the Fourier transform or the wavelet transform. The frequency components are then used, for example, to analyze or manipulate the input signal in the frequency domain.
One problem with traditional transforms is that they operate over a fixed function base that presumes the linearity and stationarity of the input signal. By way of example, the Fourier transform presumes that the input signal is composed of a summation of simple, sinusoidal functions of varying frequencies. In many cases, this presumption is incorrect. For instance, several types of real-world (i.e., physical) signals, such as time series data collected from physiological phenomena, environmental phenomena, and economic systems are nonlinear and/or nonstationary in nature. As a result, traditional transform techniques such as the Fourier transform and wavelet transform are poorly suited for analyzing these types of signals.
Recent research has led to the development of data analysis methods that overcome some of the deficiencies of fixed function base transforms. One such method, known as the Hilbert Huang Transform (HHT), relies on an adaptive function base to more accurately analyze nonlinear and nonstationary signals. In particular, HHT utilizes a decomposition technique known as Empirical Mode Decomposition (EMD) to decompose an input signal into a series of component functions called Intrinsic Mode Functions (IMFs). Each IMF has a meaningful instantaneous frequency, thereby providing a more accurate physical representation of an input signal that may be nonlinear and/or nonstationary. A detailed discussion of the Hilbert Huang Transform, Empirical Mode Decomposition, and Intrinsic Mode Functions may be found in [Huang, Norden et al., The Empirical Mode Decomposition and the Hilbert Spectrum for Nonlinear and Nonstationary Time Series Analysis, Proceedings of the Royal Society of London, A (1998) v. 454, 903-995], which is incorporated herein in its entirety for all purposes.
Although HHT represents an improvement over traditional transform techniques, its corresponding EMD method still had several shortcomings. For example, EMD requires a “sifting” process to approximate the upper and lower envelopes of an input signal. This sifting involves repetitively fitting a cubic spline to the maxima and minima of the input signal until a stopping condition is reached, which can be very time-consuming. Further, since the fitted cubic spline is an approximated, rather than true, envelope of the input function, some overshoot or undershoot may be introduced by the spline interpolation, resulting in an IMF that may not have fully symmetric envelopes. Yet further, selecting the stopping condition for the sifting process is a subjective determination that may result in different IMFs for the same input signal.